A) Householder's method B) Raphson's approximation C) Newton's method D) Halley's method
A) Isaac Newton and Joseph Raphson B) Joseph Raphson and Carl Friedrich Gauss C) Isaac Newton and Leonhard Euler D) Isaac Newton and Al-Khwarizmi
A) To find the maximum value of a function B) To find a root of a function C) To calculate the derivative of a function D) To integrate a complex function
A) Larger derivatives of a function B) Linear equations for curve fitting C) Exact solutions without iteration D) Better approximations to the roots of a function
A) A step function only B) A purely constant function C) An imaginary-valued function D) A real-valued function
A) Its integral and two boundary points B) Its derivative and an initial guess C) Its second derivative and a final value D) An exact root and a tolerance level
A) The y-intercept of the secant line at x0 B) The inflection point of the curve C) The x-intercept of the tangent line at (x0, f(x0)) D) The local minimum of the function near x0
A) The y-intercept of the linear approximation B) The unique root of the linear approximation C) The slope of the linear approximation D) The maximum point of the linear approximation
A) It roughly doubles B) It increases by a constant factor of two C) It increases linearly D) It triples with every step
A) Monte Carlo methods B) Euler's methods C) Householder's methods D) Runge-Kutta methods
A) Gradient descent B) Bisection method C) Halley's method D) Secant method
A) Only polynomial functions of degree two B) Complex functions and systems of equations C) Discrete probability distributions D) Only differential equations
A) A horizontal line at f(xn) B) The secant line connecting the endpoints C) The vertical asymptote D) The tangent line to the curve
A) f'(xn) cannot equal zero B) f'(xn) must be a complex number C) f'(xn) must equal one D) f'(xn) must equal zero
A) It diverges immediately B) It results in a division by zero C) It requires an infinite number of steps D) It generally requires fewer iterations to converge
A) Cubic B) Linear C) Logarithmic D) At least quadratic
A) They always diverge on real functions B) They require no derivatives at all C) They cannot be used on computers D) Extra computations required for each step
A) If the initial guess is too close to the root B) If the computer has too much memory C) If f or its derivatives are computationally expensive to evaluate D) If the function has too many roots
A) The ratio of f(xn) to f'(xn) B) The sum of f(xn) and f'(xn) C) The derivative divided by the function itself D) The product of f(xn) and f'(xn)
A) The exact final answer B) The convergence threshold C) The absolute maximum D) An initial guess
A) Renaissance period B) Golden Age of Athens C) Victorian era D) Old Babylonian period
A) 1st–2nd century CE B) 9th–11th century CE C) 16th–17th century CE D) 19th–16th century BCE
A) 1st–2nd century CE B) 17th–18th century CE C) 19th–16th century BCE D) 9th–10th century CE
A) Jamshīd al-Kāshī B) Seki Kōwa C) Sharaf al-Dīn al-Ṭūsī D) François Viète
A) 1711 B) 1680 C) 1633 D) 1427
A) 973–1048 B) 1540–1603 C) 1135–1213 D) 1840–1900
A) 1135–1213 B) 1633–1711 C) 1540–1603 D) 973–1048
A) Thomas Simpson B) Arthur Cayley C) John Wallis D) Henry Briggs
A) William Jones B) Joseph Raphson C) François Viète D) Seki Kōwa
A) 1540–1603 B) 1648–1725 C) 1135–1213 D) 1821–1895
A) John Colson B) Henry Briggs C) Arthur Cayley D) Thomas Simpson
A) 1669 B) 1687 C) 1711 D) 1671
A) William Jones B) John Wallis C) John Colson D) Joseph Raphson
A) 1671 B) 1690 C) 1685 D) 1669
A) William Jones B) John Colson C) Thomas Simpson D) John Wallis
A) Polynomials B) Trigonometric equations C) Differential equations D) Logarithmic equations
A) Complex roots B) Error corrections C) Optimization factors D) Derivative gradients
A) Constant terms B) Higher-degree terms C) The linear coefficients D) Initial root estimates
A) Binomial series B) Taylor series C) Fourier series D) Geometric series
A) Euler's equation B) Heat equation C) Kepler's equation D) Wave equation
A) Miftāḥ al-Ḥisāb B) Philosophiæ Naturalis Principia Mathematica C) De analysi per aequationes D) Method of Fluxions
A) 1687 B) 1711 C) 1740 D) 1633
A) Seki Kōwa B) François Viète C) Jamshīd al-Kāshī D) Sharaf al-Dīn al-Ṭūsī
A) The use of polynomials B) The use of error corrections C) The connection with calculus D) An initial estimate
A) John Wallis B) Joseph Raphson C) Thomas Simpson D) William Jones
A) The derivative function B) The original polynomial C) Higher-degree terms D) Taylor series expansion
A) A complex root formula B) A geometric optimization model C) A reusable iterative expression D) A nonpolynomial Taylor series
A) 1740 B) 1711 C) 1690 D) 1879
A) Approximating square areas B) Extracting error corrections C) Finding roots of polynomials D) Solving optimization problems
A) Arthur Cayley B) John Wallis C) Joseph Raphson D) Thomas Simpson
A) The same B) Slower C) Faster D) Quadratic
A) Removing the derivative B) Decreasing the number of iterations C) Successive over-relaxation D) Linear interpolation
A) The derivative B) The multiplicity C) The solution D) The initial error
A) x / m B) m / x C) xm / m D) m * x
A) Infinitely many iterations B) A single iteration C) Two iterations D) Ten iterations
A) Only for negative values B) Only for positive values C) Yes D) No
A) 1/2 x_n B) 1/4 x_n C) x_n2 D) 2 x_n
A) Dozens B) Exactly two C) Thousands D) Only one
A) Quadratic B) Superlinear but subquadratic C) Cubic D) Purely linear
A) About 2 B) About 10 C) About 1/2 D) About 4/3
A) Increasing rapidly B) Decreasing slowly C) Oscillating wildly D) Converging instantly
A) The 200th iterate B) The 2nd iterate C) The 10th iterate D) The 50th iterate
A) Convergent behavior B) Linear behavior C) Undefined behavior D) Oscillatory behavior
A) 0.99000 B) 0.06317 C) 1.00000 D) -1.76929
A) 1.00196 B) 1.00628 C) 0.03651 D) -0.06317
A) Oscillation stops permanently B) The method always converges faster C) Iterates can be sent outside the domain D) The derivative becomes zero everywhere
A) Constant B) Local C) Universal D) Global
A) Taylor's theorem B) Mean value theorem C) Pythagorean theorem D) Intermediate value theorem
A) α - xi B) xi / α C) xi - α D) xi + α
A) Undefined B) Zero C) Negative D) Positive
A) Undefined B) Positive C) Negative D) Zero
A) A matrix B) A vector C) A scalar D) A tensor
A) Calculus integration B) Matrix inversion C) Trigonometric operations D) Basic arithmetic operations
A) 306.5 B) 305.5 C) 612.5 D) 613.0
A) 2 decimal places B) 5 decimal places C) 10 decimal places D) 12 decimal places
A) 2mm B) 10mm C) 1mm D) 5mm
A) Maximum volume B) Minimum height C) Maximum thickness D) Minimum surface area
A) 1000 B) 10000 C) 1000000 D) 100000000
A) h + b + w = 1136000 B) 2hbW = 1136000 C) hbw = 2000000 D) hbw = 1136000
A) 1136000 / (hb) B) 1136000 / w C) 1136000 / (h + b) D) 1136000 * hb
A) [w, h] B) [w, b] C) [b, w, h] D) [b, h]
A) [50, 50] B) [0, 0] C) [1, 1] D) [100, 100]
A) [50.123, 110.456] B) [138.5684895, 65.67176536] C) [100, 100] D) [65.67176536, 138.5684895]
A) Jacobian matrix B) Hessian matrix C) Identity matrix D) Vector matrix
A) Zeroes B) Asymptotes C) Integrals D) Derivatives
A) Polygons B) Straight lines C) Circles D) Fractals
A) Jürgen Moser B) Richard Hamilton C) John Nash D) Curt McMullen
A) Euclidean space B) Hilbert space C) Fréchet space D) Banach space
A) Strictly positive B) Continuously differentiable C) Boundedly invertible D) Symmetric
A) 1970s B) 1950s C) 1960s D) 1940s
A) Complex initial conditions B) Universal starting points C) Fréchet derivatives D) Smoothing operators
A) 1950s B) 1980s C) 1970s D) 1960s
A) Quantum mechanics B) Complex analysis C) Fluid dynamics D) Celestial mechanics
A) Curt McMullen B) Jürgen Moser C) John Nash D) Richard Hamilton
A) Banach spaces B) Euclidean spaces C) Fréchet spaces D) Hilbert spaces
A) First-order B) Fourth-order C) Cubic D) Second-order
A) Newton B) Maehly C) Pafnuty Chebyshev D) Hensel
A) Hensel's lemma B) Quasi-Newton method C) Maehly's procedure D) Chebyshev's method
A) The entire field B) The unit ball C) The set of integers D) The set of primes
A) The p-adic analog B) The Maehly analog C) The q-analog D) The Chebyshev analog |